Question: The graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has a horizontal asymptote at $y=a$.  What is $a$?
Solution: To determine the horizontal asymptotes, we consider what happens as $x$ becomes very large.  It appears that, as $x$ becomes very large, the rational function becomes more and more like  \[y\approx\frac{5x^2}{3x^2},\]so it should become closer and closer to $\frac53$.

We can see this explicitly by dividing both the numerator and denominator by $x^2$.  This gives  \[y=\frac{5-\frac{9}{x^2}}{3+\frac{5}{x}+\frac{2}{x^2}}.\]Indeed, as $x$ gets larger, all of the terms other than 5 in the numerator and 3 in the denominator become very small, so the horizontal asymptote is $y=\boxed{\frac53}$.